A MOVING BOUNDARY VALUE PROBLEM IN SOFT TISSUE MECHANICS
A MOVING BOUNDARY VALUE PROBLEM IN SOFT TISSUE MECHANICS
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通讯作者：
M. Stastna
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作者：
M. Stastna
软物质（如生物组织）的力学理论通常在两种理论之间徘徊，一种理论可以代表实验观察到的行为的最大范围，另一种理论可以以合理有效的方式给出解决方案。虽然廉价、高速计算的出现扩大了可用于给定情况的数值分析的理论范围，但软件验证、实验解释和机械参数确定等问题仍然需要一套简单、易于解释、易于分析的理论。在下面，我们考虑具有移动边界的理想配置中的线性固结理论。利用可逆坐标变换，导出了一个固定定义域上的非线性问题。对于稳态自由边值问题，在稳定极限下，通过逆变换，导出了固定定义域上的解析解。我们采用标准数值方法证明了随时间变化的非线性问题趋于稳态极限，并讨论了演化到稳态的速率取决于物理参数。随后，利用稳态结果构建了一种求解系留管几何结构稳态固结自由边值问题的算法。最后，该算法的实现讨论了几个案例研究，这些案例研究概述了参数空间中自由边界影响显著的部分。
Mechanical theories of soft matter such as biological tissues often tread a fine line between a theory that can represent the maximal range of experimentally observed behaviour and one that can yield solutions in a reasonably efficient manner. While the advent of cheap, high speed computation has extended the range of theories that can be used for numerical analysis of a given situation, questions of software validation, interpretation of experiment and the determination of mechanical parameters still require a set of simple to interpret, analytically tractable theories. In the following we consider the linear theory of consolidation in an idealized configuration with a moving boundary. Employing an invertible coordinate transformation we derive a nonlinear problem on a fixed domain. In the steady limit we derive analytical solutions on the fixed domain, and by inverting the transformation, for the steady state free boundary value problem. We employ standard numerical methods to show that the time-dependent nonlinear problem tends to the steady state limit and discuss the rate at which the evolution to the steady state depends on the physical parameters. The steady state results are subsequently utilized to construct an algorithm for solving the free boundary value problem for steady state consolidation in a tethered tube geometry. Finally, the algorithm is implemented to discuss several case studies which outline the portions of parameter space in which the effect of the free boundary is significant.